Subgroups — C₂×#20295504 #28376223

Order 13952.

1488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295504	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₀₉	C₂	1	8	yes	maximal, complemented
3	C₂×#16812658	C₂	1	6	yes	maximal
4	C₂³×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₁₀₉	C₂²	1	64	yes	complemented
6	#16812658	C₂²	1	48	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	12	yes	—
8	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	12	yes	—
10	C₂²×Q₈×C₁₀₉	C₄	1	16	yes	complemented
11	C₂²×Q₈×C₁₀₉	C₂²	1	4	yes	—
12	C₂²×Q₈×C₁₀₉	C₂²	1	8	yes	—
13	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
14	C₂²×Dic₁₀₉	C₂³	1	8	yes	—
15	C₄⋊Dic₁₀₉	C₂³	1	48	yes	—
16	C₂²⋊Dic₁₀₉	C₂³	1	16	yes	—
17	C₄×Dic₁₀₉	C₂³	1	48	yes	—
18	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₁₀₉	C₂×C₄	1	32	yes	—
20	C₂×Q₈×C₁₀₉	C₂×C₄	1	64	yes	complemented
21	C₂×Q₈×C₁₀₉	C₂³	1	16	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	24	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
25	C₂³×C₁₀₉	—	2	4	no	—
26	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
27	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
28	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
29	Q₈×C₁₀₉	C₂²×C₄	1	64	yes	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	24	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	48	yes	—
32	C₂×C₄×C₁₀₉	C₂⁴	1	12	yes	—
33	C₂×Dic₁₀₉	—	2	16	no	—
34	C₂×Dic₁₀₉	C₂⁴	1	16	yes	—
35	C₂²×C₁₀₉	—	2	8	no	—
36	C₂²×C₁₀₉	—	2	4	no	—
37	C₂²×C₁₀₉	C₂³×C₄	1	4	yes	—
38	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
39	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
40	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂³×C₄	1	24	yes	—
42	Dic₁₀₉	—	2	16	no	—
43	C₂×C₁₀₉	—	2	4	no	—
44	C₂×C₁₀₉	#519	1	4	yes	complemented
45	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	2	yes	—
47	C₂×#519	—	109	1	no	maximal, Sylow
48	C₁₀₉	C₂×#519	1	1	yes	characteristic, complemented, Sylow
49	C₂³×Q₈	D₁₀₉	1	1	yes	characteristic
50	#519	—	109	16	no	—
51	C₂×C₄×Q₈	—	109	8	no	—
52	C₂×#117	—	109	6	no	—
53	C₂²×Q₈	Dic₁₀₉	1	16	yes	complemented
54	C₂²×Q₈	C₂×D₁₀₉	1	4	yes	—
55	C₂²×Q₈	C₂×D₁₀₉	1	8	yes	—
56	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
57	C₄×Q₈	—	109	64	no	—
58	#117	—	109	48	no	—
59	C₂×C₄⋊C₄	—	109	12	no	—
60	C₂×C₂²⋊C₄	—	109	4	no	—
61	C₂×C₄²	—	109	12	no	—
62	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
63	C₂×Q₈	C₂×Dic₁₀₉	1	32	yes	—
64	C₂×Q₈	C₂×Dic₁₀₉	1	64	yes	complemented
65	C₂×Q₈	C₂²×D₁₀₉	1	16	yes	—
66	C₂²×C₄	—	109	8	no	—
67	C₂²×C₄	C₂×Dic₁₀₉	1	24	yes	—
68	C₂²×C₄	C₂²×D₁₀₉	1	12	yes	—
69	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
70	C₄⋊C₄	—	109	48	no	—
71	C₂²⋊C₄	—	109	16	no	—
72	C₄²	—	109	48	no	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
75	C₂³	C₂³×D₁₀₉	1	2	yes	—
76	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
77	Q₈	C₂²×Dic₁₀₉	1	64	yes	—
78	C₂×C₄	—	109	16	no	—
79	C₂×C₄	—	218	16	no	—
80	C₂×C₄	C₂²×Dic₁₀₉	1	24	yes	—
81	C₂×C₄	C₂²×Dic₁₀₉	1	48	yes	—
82	C₂×C₄	C₂³×D₁₀₉	1	12	yes	—
83	C₂²	—	2	8	no	—
84	C₂²	—	2	4	no	—
85	C₂²	C₂³×Dic₁₀₉	1	4	yes	—
86	C₂²	C₂³×Dic₁₀₉	1	2	yes	—
87	C₂²	#16812747	1	4	yes	—
88	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
89	C₄	—	218	16	no	—
90	C₄	C₂³×Dic₁₀₉	1	24	yes	—
91	C₂	—	2	4	no	—
92	C₂	#20295504	1	4	yes	direct product factor
93	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic
94	C₂	C₂×#16812747	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂³×Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×C₁₀₉	C₂³
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂³×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#519
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295504	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₀₉	C₂	semidirect
5. Q₈×Dic₁₀₉	C₂²	semidirect
10. C₂²×Q₈×C₁₀₉	C₄	semidirect
20. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
44. C₂×C₁₀₉	#519	semidirect
48. C₁₀₉	C₂×#519	semidirect
53. C₂²×Q₈	Dic₁₀₉	semidirect
64. C₂×Q₈	C₂×Dic₁₀₉	semidirect
92. C₂	#20295504	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295504	2
C₂×Q₈×Dic₁₀₉	2
C₂×#16812658	2
C₂³×Q₈×C₁₀₉	2
C₂×#519	109

Subgroups — C2×#20295504 #28376223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295504 #28376223